Propagation of guided waves through weak penetrable scatterers

Propagation of guided waves through weak penetrable scatterers

Agnes Maurela)

Institut Langevin/LOA, UMR 7587, CNRS-ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France

Jean-Francois MercierPoems, UMR 7231, CNRS-ENSTA-INRIA, 32 bld Victor, Paris 75015, France

(Received 26 August 2011; revised 28 December 2011; accepted 5 January 2012)

The scattering of a scalar wave propagating in a waveguide containing weak penetrable scatterers

is inspected in the Born approximation. The scatterers are of arbitrary shape and present a contrast

both in density and in wavespeed (or bulk modulus), a situation that can be translated in the context

of SH waves, water waves, or transverse electric/transverse magnetic polarized electromagnetic

waves. For small size inclusions compared to the waveguide height, analytical expressions of the

transmission and reflection coefficients are derived, and compared to results of direct numerical

simulations. The cases of periodically and randomly distributed inclusions are considered in more

detail, and compared with unbounded propagation through inclusions. Comparisons with previous

results valid in the low frequency regime are proposed. VC 2012 Acoustical Society of America.

[DOI: 10.1121/1.3682037]

PACS number(s): 43.20.Fn, 43.20.Mv, 43.20.Bi [ANN] Pages: 1874–1889

I. INTRODUCTION

The interaction of guided waves with scatterers has been

studied in a large variety of physical contexts. In the context of

dielectric waveguides, as used in optical communication, inter-

est lies in the understanding of the effects of impurities due to

material defects and in the possibility of utilizing deliberate iso-

lated impurities as passive optical components.1–3 The detec-

tion and the characterization of buried defects or targets

motivates research in the context of acoustic waves,4–6 water

waves,7,8 and in the context of elastic waves.9,10 The problem

of scattering by a single scatterer centered in a waveguide is

analogous to the scattering problem by a periodic array of scat-

terers. This latter problem has its own interest and it has been

extensively studied, notably to handle for several mathematical

difficulties and to develop accurate and efficient numerical

methods.11–13 Also, a recent increasing interest for these peri-

odic structures is inspired by the success of studies of photonic

crystals14 and possible applications in other contexts of waves.

The literature is vast, see Refs. 15 and 16 for recent works

using a supercell approach and for a review, also see Ref. 17.

The scattering of waves by obstacles, either penetrable or

not, is a complicated problem and exact analytical solutions are

in general unavailable. This is why it is of common practice to

employ approximation methods, for their mathematical sim-

plicity and applicability to any shape and structure, also for the

easy implementation when considering inverse problems.

Among the approximate methods, the Born approximation

(Debye–Ganz or Debye–Rayleigh approximation in the context

of electromagnetic waves) is the most commonly used. See

Refs. 18–21 regarding imaging techniques in the context of

guided waves. This approximation employs the incident field

as the total field in its first iteration, an approximation expected

to be valid for weak scattering, namely predominant single

scattering contribution [then, the (nþ 1)th iteration employs

the field at the nth iteration as the total field]. Extensive litera-

ture exists on the discussion of the validity of the Born approxi-

mation. Some studies focus on theoretical aspects: Does the

Born series converge and if so, how large is the error when bro-

ken off after a finite number of terms? Other studies propose a

heuristic criterion for its range of validity by comparison of

the approximated solution with an exact analytical result,

when available, or with experiments/numerical simulations.

We refer to several studies in the context of acoustic22–25

electromagnetism26–28 and quantum mechanics.29,30 Because of

weak scattering approximation, the range of validity concerns

the low frequency regime when impenetrable inclusions are

considered (sound soft/Dirichlet or sound hard/Neumann scat-

terers). Less attention has been paid to penetrable inclusions/

scatterers, where the scattering strength results from a combina-

tion of the scatterer size and of the strength of the contrasts in

density and bulk modulus for acoustic waves, ocean depth in

the context of water waves, permittivity and permeability for

electromagnetic waves, etc. We mention the experimental

study presented in Ref. 31 on light scattering by penetrable

blood cells with low contrast in the refractive index showing

that the Born approximation is accurate well beyond the low

frequency regime (until kR� 10–100, where k is the wavenum-

ber and R is the blood cell size). This latter situation corre-

sponds to weak scattering because of the weak contrast in the

optical index between the cell and the surrounding aqueous me-

dium rather than because of small scatterer size. Finally, of par-

ticular interest is Ref. 32, where an experimental method is

presented to isolate in an experimental signal the contribution

of single scattering from the multiple scattering contribution,

offering an experimental test for the first Born approximation.

The present paper addresses the problem of acoustic i.e.,

scalar waves propagating in a waveguide containing a single

inclusion or a set of penetrable inclusions. In the context of

acoustic waves in fluid with inclusions having contrasts both

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

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in density q and in bulk modulus B, the corresponding equa-

tion for the pressure p is

r � 1

qðrÞrpðrÞ� �

þ x2

BðrÞ pðrÞ ¼ 0; (1)

with continuity of p and the normal component of rp=q at

the inclusion boundaries. This equation applies in other con-

texts of scalar waves: (i) The outplane displacement u of SH

waves in solids

r � ðlðrÞruðrÞÞ þ qðrÞx2 uðrÞ ¼ 0

with (l, q) the shear modulus and the mass density of the

elastic medium, (ii) the surface elevation g for water (grav-

ity) waves propagating in varying shallow water depth h(r)

(with g the acceleration due to gravity)

r � ðhðrÞrgðrÞÞ þ x2

ggðrÞ ¼ 0;

and (iii) for the electric field E or the magnetizing field H for

polarized transverse magnetic or transverse electric electro-

magnetic waves, with permittivity � and permeability l

r 1

lðrÞrEðrÞ� �

þ x2�ðrÞEðrÞ ¼ 0;

r 1

�ðrÞrHðrÞ� �

þ x2l rð ÞH rð Þ ¼ 0:

The paper is organized as follows. In Sec. II, an integral rep-

resentation for the pn(x)-coefficients of the modal decompo-

sition of the solution p(r) is derived. The resulting Eq. (8) is

the basic equation to implement numerical calculations in

the successive Born approximations pNBC(x, y) and examples

for inclusion of nontrivial shapes are given. It can be used

also to derive analytical expressions of the transmission and

reflection coefficients [the corresponding field is pB(x, y)].

Notably, simplifications are presented when considering

inclusions of any shape, but of size small compared to the

waveguide height [Eq. (16), note that this assumption does

not restrict the solution to the low frequency regime].

Section III presents a discussion on the validity of the

predictions pNBC and pB, the reference calculations being

performed using the finite element MELINA code.33 The scat-

tering properties are dependent on multiple parameters that

are discussed. Basically, and as expected in this iterative

approach, the relevant parameter to predict the accuracy of

our analytical pB and numerical pNBC predictions is the am-

plitude of the scattered field, that results from a combination

of the inclusion size and of the contrasts in wavespeed and

in density. Our conclusion is that for an incident wave field

of unit amplitude and denoting �s the amplitude of the scat-

tered field ps, our estimate psB is accurate up to �2

s for �s until

values close (but smaller) than 100%. When this is the case,

iterating NBC causes the solution to converge.

Finally, Sec. IV focuses on the properties of the scat-

tering by periodic and random distributions of inclusions.

In the former case, the comparison with the scattering by

an unbounded infinite array of inclusions allows us to

recover the usual grating law and in connection, the

Wood’s anomaly. Preliminary numerical experiments

(much more difficult because numerical averages are

needed) are presented for random distribution of inclusions.

Reasonable agreement with our prediction is found. In both

periodic and random configurations, comparisons with per-

turbative approaches in the low frequency regime are

proposed.

II. INTEGRAL REPRESENTATION OF THE FIELDSCATTERED BY INCLUSIONS

In this section, we use the wave equation in an integral

form together with a modal decomposition onto the trans-

verse modes of the waveguide. Accounting for the appropri-

ate boundary conditions at the inclusion boundaries leads to

an integral representation, Eq. (8), of the modal coefficients

pn(x). For weak scattering strength of the inclusions, an

approximated solution is proposed using the Born approxi-

mation. This can be done numerically by iterating the

successive Born solutions [Eq. (11)]. Alternatively, an ana-

lytical solution is proposed when assuming that the size of

the inclusions is small compared to the height of the wave-

guide. Note that this latter assumption does not assume

a priori a small inclusion size compared to the incident

wavelength.

A. Description of the problem

The problem under consideration is shown in Fig. 1: An

acoustic waveguide in the two-dimensional (2D) -space

r¼ (x, y), with wavespeed c and mass density q contains

inclusions that presents a contrast in density and in wave-

speed with respect to the host medium ~q; ~cð Þ. The inclusions

are located within a region 0< x< L.A wave is generated by a source located at x¼ xs< 0

[encapsulated in the source term s(r), this choice, rather than

a plane wave coming from �1 will be justified later] and

propagates in the waveguide. In the x-frequency regime, the

wave satisfies the wave equation

r � 1

qðrÞrpðrÞ� �

þ x2

B rð Þ pðrÞ ¼ s rð Þ; (2)

in the waveguide (domain X) where q(r)¼q and B(r)

¼B: qc2 and inside the inclusion (domain ~X) where

FIG. 1. Ensemble of penetrable inclusions in a waveguide and parametriza-

tion of the inclusion boundary.

J. Acoust. Soc. Am., Vol. 131, No. 3, March 2012 A. Maurel and J.-F. Mercier: Propagation of guided waves through scatterers 1875


q rð Þ ¼ ~q and B rð Þ ¼ ~B � ~q~c2. We also define the wavenum-

ber k:x/c. The boundary conditions on the walls of the

waveguides at y¼ 0, H are

@ypðx; 0Þ ¼ @ypðx;HÞ ¼ 0; (3)

and we have continuity of p and the normal component of

rp/q at the boundary of the inclusion. The source term s(x,

y) is written as

sðx; yÞ ¼ 1

qdðx� xsÞwNðyÞ;

where wn(y) are the usual eigenfunctions of the waveguide

wnðyÞ �ffiffiffi2p

cos npyH

w0ðyÞ ¼ 1;

�

satisfying @ywn(0)¼ @yw(H)¼ 0 on the boundaries of

the waveguide and satisfying the orthogonality relationÐH0

dy wn yð Þwm yð Þ ¼ Hdnm. These wn-functions form a basis

of H1(0, H) (the function and its first derivative are square-

integrable).

B. Modal decomposition and derivation of the integralrepresentation

Before deriving the modal decomposition of the pres-

sure, let us write problem (2) in a variational form: By multi-

plying Eq. (2) by a test function q(r)¼Q(x)wn(y), with Q(x)

compactly supported, and integrating over the whole domain

of the waveguide Xt ¼ X [ ~X, we get

ðXt

dr

�rpðrÞ � rqðrÞ � k2pðrÞqðrÞ

�

þð

~Xdr½ðq=~q� 1ÞrpðrÞ � rqðrÞ

� k2ðB= ~B� 1ÞpðrÞqðrÞ� ¼ �qð

Xt

dr sðrÞqðrÞ: (4)

We now expand p on the wn

pðrÞ ¼X

m

pmðxÞwmðyÞ: (5)

The modal coefficients of rp are obtained accounting for

the continuity of p at the boundary of the inclusion

ðH

0

@xpðx; yÞwmðyÞ ¼d

dx

ðH

0

dy pðx; yÞwmðyÞ ¼ Hp0mðxÞðH

0

dy @ypðx; yÞw0mðyÞ ¼ �ðH

0

dy pðx; yÞw00mðyÞ ¼mpH

� 2

HpmðxÞ;

8>><>>:

so that rp can be expended on the wm-functions (respectively, w0m-functions)

rpðrÞ ¼X

m

p0mðxÞwmðyÞpmðxÞw0mðyÞ

� �:

Note that we have used the fact that the w0m yð Þ-functions also form a basis withÐH

0w0m yð Þw0n yð Þ ¼ mp=Hð Þ2dnm. Equation (4)

becomes, using the orthogonality relations of the wm and w0m functions, the variational formulation

ðþ1�1

dx

�p0n xð ÞQ0 xð Þ � k2

npn xð ÞQ xð Þ�þ 1

Hq=~q� 1ð Þ

Xm

ðþ1�1

dx Cnm xð Þp0m xð ÞQ0 xð Þ þ Dnm xð Þpm xð ÞQ xð Þ �

� k2

HB= ~B� 1� X

m

ðþ1�1

dx Cnm xð Þpm xð ÞQ xð Þ ¼ �ðþ1�1

dx Q xð Þd x� xsð ÞdnN;

where we have defined

k2n � k2 � np

H

� 2

Cnm xð Þ �ðb xð Þ

a xð Þdy wm yð Þwn yð Þ ¼ anm

2Y sinc

n� mð ÞpY

Hþ Ysinc

n� mð ÞpY

H

� �Y¼b xð Þ

Y¼a xð Þ

Dnm xð Þ �ðb xð Þ

a xð Þdy w0m yð Þw0n yð Þ ¼ nm

p2

H2Y sinc

n� mð ÞpY

Hþ Ysinc

nþ mð ÞpY

H

� �Y¼b xð Þ

Y¼a xð Þ

8>>>>>>><>>>>>>>:

(6)

for x 2 Cx otherwise Cnm(x)¼Dnm(x)¼ 0 (for x 62 Cx), with

Cx the x-range containing one or more inclusions (Fig. 1).

We have defined the function sinc x: sin x/x, and the coeffi-

cients a00 ¼ 1, an0 ¼ a0n ¼ffiffiffi2p

for n 6¼ 0 and anm ¼ 2 for

m 6¼ 0, n 6¼ 0. In the previous expression, (x, a(x)) and

(x, b(x)) correspond to the parameterization of the interface

1876 J. Acoust. Soc. Am., Vol. 131, No. 3, March 2012 A. Maurel and J.-F. Mercier: Propagation of guided waves through scatterers


between one inclusion and the host medium (Fig. 1). In the

case of several inclusions occupying ~X ¼ [~Xi, the result

comes from linearity as a sum over all the inclusions: in

Eq. (4),Ð

~X can be replaced byP

i

Ð~X . Integrating the above-

mentioned variational formulation by parts, we obtain

p00n xð Þ þ k2npn xð Þ

¼ q=~q� 1ð ÞH

Xm

� Cnm xð Þp0m xð Þ� 0h

þ Dnm xð Þpm xð Þ� �k2 B= ~B� 1�

H

Xm

Cnm xð Þpm xð Þ

þ d x� xsð ÞdnN (7)

owing to the fact that Cnm(x) and Q(x) vanish at 61. The

one-dimensional (1D) Green’s function

gn xð Þ ¼ eikn xj j

2ikn

satisfying

@2

@x2þ k2

n

� �gn x� x0ð Þ ¼ d x� x0ð Þ

is used to get an integral representation of the solution pn(x)

of Eq. (7)

pn xð Þ ¼ gN x� xsð ÞdnN �k2 B= ~B� 1�

H

Xm

ðcx

dx0gn x� x0ð ÞCnm x0ð Þpm x0ð Þ

þ q=~q� 1ð ÞH

Xm

ðcx

dx0 �g0n x� x0ð ÞCnm x0ð Þp0m x0ð Þ þ gn x� x0ð ÞDnm x0ð Þpm x0ð Þ �

; (8)

where g0n x� x0ð Þ � @xgn x� x0ð Þ. Note that the dependence

on the solution pn(x) on the shape and position of

the inclusion(s) is encapsulated in the coefficients Cnm and

Dnm. Incidentally, the integral representation of Eq. (7)

causes a term at the boundaries x0 ! 61 of the form

@x0gn x� x0ð Þpn x0ð Þ � gn x� x0ð Þp0n x0ð Þ �

x0!61 to appear.

Although Green’s function of the 1D problem does not van-

ish at infinity, this term vanishes because of the form of our

source term: with a source located at xs, we have

@x0 fn x0ð Þ ¼ 6iknfn x0ð Þ for both fn ¼ gn or pn when x0 ! 61.

This property causes the term at the boundaries to vanish.

In practice, we can choose the position of the source xs

outside the region containing the inclusions, say xs< 0.

Thus, the incident wave can be written (for xs< x<þ1)

pinc x; yð Þ � gN x� xsð ÞwN yð Þ ¼ AeikNxwN yð Þ; (9)

which corresponds to a particular right going guided mode

(mode N). In the following, without loss of generality, we

choose A ¼ 1.

We inspect in the following section the possibility to

evaluate pn(x) in the Born approximation.

C. Iterative solutions using the Born approximations

The first iteration in the Born approximation consists

of identifying the total field with the incident field, leading

to

p0n xð Þ ¼ eikNxdNn; (10)

The expression of pn(x) in Eq. (8) can be evaluated perturba-

tively in the successive Born approximation, namely

piþ1n xð Þ ¼ eikNxdnN þ

ðq=~q� 1ÞH

Xm

ðCx

dx0 �g0n x� x0ð ÞCnm x0ð Þpim0 x0ð Þ þ gn x� x0ð ÞDnm x0ð Þpi

m x0ð Þ �

�k2 B= ~B� 1�

H

Xm

ðCx

dx0 gn x� x0ð ÞCnm x0ð Þpim x0ð Þ; (11)

where pin denotes the solution in the ith-Born

approximation.

Numerically, pin xð Þ can be evaluated for any inclusion

shape by truncating the sums over m to some Nmod values

(in general, at least the propagating modes are considered,

the evanescent modes being relevant in the near field

only), for i¼ 1 to some Nit order. The corresponding 2D

solution pNBC(x, y) (NBC stands for numerical Born calcu-

lation) is

pNBC x; yð Þ ¼XNmod

n¼0

pNitn xð Þwn yð Þ; (12)

and it depends on both the number of considered modes

Nmod and on the order of the Born approximation Nit. A sim-

ple implementation can be found in the Appendix.

In the following, we give typical examples of the NBC for

nontrivial inclusion shapes: S1 where the inclusion boundary

(xb,yb) is parametrized with t using xb/H¼ 0.175 cos tþ 2.5,



yb¼ 0.15xbþ (xb� 5)2/2þ 0.125 sin t � 0.4, and S2 where (xb,yb) is parametrized using xb/H¼ 0.175 cos tþ 2.5 and

yb¼ 0.3xb/Hþ 2(xb/H� 2.5)2þ 0.125 sin t � 0.4.

To allow for comparison, a reference solution pM(x, y) is

calculated using the finite element code MELINA.33 The modal

components pn(x) are deduced by interpolating pM(x, y) on a

regular mesh and by projecting numerically onto the wn(y)-ba-

sis. The scattered reference field psM x; yð Þ and the field

psNBC x; yð Þ calculated from Eq. (12) [see also Eqs. (A3) and

(A4)] are reported in Fig. 2, and the corresponding modal

components pn(x) in Fig. 3. We have considered an incident

plane wave of unity amplitude (for clarity of the curves of the

modal components, the modal component of the incident

wave eikNx is omitted). The maximum scattering amplitude are

respectively 0.9 and 0.3, and NBC appears to be able to cap-

ture accurately the solution (here, Nmod¼ 10 and Nit¼ 6 have

been considered).

D. Analytical expressions of the reflection andtransmission coefficients

The first Born approximation leads to simple forms of

the reflection and transmission coefficients, defined by

p1n xs < x < 0ð Þ ¼ eikNxdnN þ RnNeiknx

p1n x > Lð Þ ¼ TnNeiknx;

�(13)

by using i¼ 0 in Eqs. (10) and (11):

RnN ¼q=~q�1ð Þ2iknH

�knkNCnN kNþ knð Þ

þ DnN kNþ knð Þ��

k2 B= ~B�1� 2iknH

CnN kNþ knð Þ;

TnN ¼ dnNþq=~q�1ð Þ2iknH

knkNCnN kN� knð Þ

þ DnN kNþ knð Þ��

k2 B= ~B�1� 2iknH

CnN kN� knð Þ; (14)

where F qð Þ �ÐCx

dx0eiqx0F x0ð Þ is basically related to the Fou-

rier transform of the inclusion shape for propagative modes.

E. Analytical solution in the case of small inclusionsize compared to the waveguide height

We inspect in this section the simplifications that

occur when considering small symmetrical inclusions,

namely for inclusions of typical size R� H (at this

stage, no assumption about the kR-value is necessary).

In that case, a xð Þ ¼ y0 � H� xð Þ, b xð Þ ¼ y0 þ H� xð Þ, and

� xð Þ � � � R=H � 1. We get the following for CnN and DnN:

CnN xð Þ ¼ 2cnN=kN � xð Þ þ O �2ð Þ; cnN � anN kNH cosnpy0

Hcos

Npy0

H

DnN xð Þ ¼ 2kndnN � xð Þ þ O �2ð Þ; dnN � 2nNp2

kNHsin

npy0

Hsin

Npy0

H;

8><>: (15)

so that Eq. (2.13) becomes

TnN ¼ dnN þ i � q=~q� 1ð Þ dnN þ cnNð Þ þ k2= knkNð Þ B= ~B� 1�

cnN

�ICx

kN � knð ÞRnN ¼ i � q=~q� 1ð Þ dnN � cnNð Þ þ k2= knkNð Þ B= ~B� 1

� cnN

�ICx

kN þ knð Þ

�(16)

FIG. 2. (Color online) Scattered field psM x; yð Þ (top), ps

NBC x; yð Þ (middle, calculated for Nmod¼ 10 and Nit¼ 6) and psB x; yð Þ [bottom, calculated from Eq. (14)]

for (left) the inclusion shape S1 and (right) the inclusion shape S2. No contrast in mass density has been considered in both cases, the contrast in wavespeed is

c=~c ¼ 1:2 and kH¼ 11.12. The errors are, for S1, ErB¼ 9%, ErNBC¼ 1.6%, with �s¼ 90% and, for S2, ErB¼ 1.1%, ErNBC¼ 0.06% with �s¼ 30%.



with

ICxqð Þ � 1

H

ðCx

dx0 � x0ð Þeiqx0 : (17)

The result is O �3ð Þ if we assume that the size of the inclusion

is of same order in both direction x and y (otherwise,

O �3ð ÞO Rx=Hð Þ with Rx the typical size along x).

The dependence of TnN and RnN on the inclusion y0-posi-

tion is encapsulated in cnN and dnN, whereas the dependence

on the x-position and on the shape of the inclusion is encap-

sulated in ICx. This latter integral can be evaluated for sev-

eral inclusion shapes. As an example, we have, for a

cylinder of radius R

ICxqð Þ ¼ pR2

H2

J1 qRð ÞqR

eiqx0 ; (18)

where Jn denotes the first kind Bessel function.

As other examples, the case of rectangular inclusions, of

lateral extension 2Rx and of height 2R gives ICxqð Þ

¼ 2 R RxA=H2ð ÞsincqRx eiqx0 or the case of a diamond shape,

of lateral extension 2A and of height 2a, corresponds to

ICxqð Þ ¼ aA=H2 sinc2 qA=2ð Þeiqx0. Note that we have not

assumed that the size of the inclusions is small with respect

to the incident wavelength 2p/kN. If the case, the wave is not

sensitive to the inclusion shape and

ICxqð Þ ¼ S

2H2eiqx0 ;

we get an additional simplification where S is the surface of

the inclusion.

III. INSPECTION OF THE VALIDITY OF THESCATTERED FIELD

In this section, we inspect the accuracies of the NBC

and of our analytical expression (referred as B) in Eq. (16).

The solution

pM x; yð Þ ¼ pinc x; yð Þ þ ps x; yð Þ; (19)

calculated using the MELINA code is assumed to be the refer-

ence solution p(x, y) and we measure the scattering strength,

for an incident wave of amplitude unity by

�s � max ps x; yð Þj j; (20)

where the maximum value is calculated over the calculation

domain. The errors are then defined as

ErB � psB � ps

�� ; ErNBC � psNBC � ps

�� ; (21)

and Xj j refers to the mean value of X over the calculation do-

main. As previously reported, we expect the Born approxi-

mations to be valid for weak scattering, namely �s � 1.

Varying the parameters kR, ~q=q, and ~c=c, we will show in

the following that

ErB ’ �2s (22)

or equivalently

ps ¼ �s ! p ¼ pinc þ psB þ �2

s (23)

for �s <�1 (up to values close to unity), meaning strong scatter-

ing regime. When this is the case, iterating the Born approxi-

mation PsNBC causes the solution to converge to ps. If not

�s > 1ð Þ, iterating the Born approximation causes the solution

to diverge. The prediction of Eq. (16) has an additional

assumption of small inclusion size compared with the height

of the waveguide 2R=H � 1, that is inspected varying 2R/H.

A. Convergence of pB, pNBC varying Nmod (Nit)

We show here the results obtained for the scattering of

an incident plane wave (in mode 0) by a cylindrical inclusion

(with, respectively, small and large 2R/H values) and a

FIG. 3. Modal components pn(x) [the modal component eikNx, here for N¼ 0,

is omitted in p0(x)] in the same cases as in Fig. 2. In solid line, psM x; yð Þ, in

dotted lines in the NBC calculations using Eq. (11) (almost indistinguishable

with the solid line) and in dashed lines in the first Born approximation

[Eq. (12)]. The curves are shifted from the zero mean for clarity.



contrast in density (respectively, contrast in wavespeed) of

0.5. In the first case, the cylindrical inclusion C1 has a

small size (kR¼ 0.55, 2R/H¼ 0.1) producing a scattered

field of amplitude �s ¼ 21%. In the second case, the cylin-

drical inclusion C2 has a large size (kR¼ 3.34, 2R/H¼ 0.6),

producing a scattered field of amplitude �s ¼ 110%. In both

cases, we have kH¼ 11.12, thus four propagative modes.

Figure 4(a) shows the variation of the error ErB as a func-

tion of the number of accounted modes and the variations

of the error ErNBC as a function of the number of modes

for increasing Nit¼ 1, 2, 3, 6, 11 iterations. We observe

ErB � ErNBC(Nit¼ 1)� 1.5% (close to �2s ¼ 4%) with no

significant variation on Nmod, starting from the four propa-

gative modes and accounting for additional evanescent

modes. Increasing Nit and Nmod causes pNBC to converge to

errors less than 0.1%. In the second case, our analytical

prediction pB is inaccurate, with an error of� 25%, see Fig.

4(b). However, it appears that pNBC has a significantly

smaller error, even for Nit¼ 1 (ErNBC¼ 10%) and rapidly

converges when increasing Nit and Nmod until less than 1%.

In that case, the large R/H value is the source of the differ-

ence between ErB and ErNBC(Nit¼ 1), as the analytical pre-

diction pB is done for small R/H-values. Also, the large

error is attributable to the high value of the scattered field,

�s close to 1.

In these two situations, the main condition to make pB

and pNBC accurate is the �s-value that clearly defined the va-

lidity condition of the Born approximation. This is inspected

further varying the other parameters that define the scattering

strength �s. Then, increasing R/H causes the analytical

prediction pB to be less and less precise [pB is accurate up to

(R/H)3, a restriction that does not affect pNBC].

B. Influence of the contrast in density or contrast inwavespeed

The contrasts in density and in wavespeed are two pa-

rameters that define the scattering strength of the inclusion.

Their influences on the accuracy of pB (close symbols) and

pNBC (open symbols for Nit¼ 1 and 6) are illustrated on the

Fig. 5, together with the amplitude �s of the scattered field.

In the range of explored contrasts (~c=c; ~q=q varying between

0.5 and 10 for an inclusion size kR¼ 0.56), the scattered

field has an amplitude �s varying between 10�2 to a value

close to 1. Within this range, the corresponding accuracy of

our analytical prediction pB roughly behaves like �2s as

expected. As previously reported, this produces a significant

error (up to 20%) for �s close to 1. We also observe that

pNBC converges when increasing Nit, for these �s<�1.

C. Influence of the inclusion size and wavenumber

As qualitatively seen in Sec. III A, considering the influ-

ence of the inclusion size has two aspects: Basically, increas-

ing R produces an increase in the scattering strength. On the

other hand, the analytical prediction pB has an additional

assumption of small R/H-values.

We report in Fig. 6(a) the errors ErB (close symbols

with dashed line) and ErNBC (open symbols with: dashed

line for Nit¼ 1 and dotted line for Nit¼ 6), together with the

value of the scattered field �s (solid line) as a function of kR.Here, the R/H value is constant and increasing kR (by

increasing k in a fixed geometry) produces an increase in the

scattering strength of the inclusion [typically ps ¼ �s / kRð Þfor a fixed geometry]. In the presented range of kR-values,

the scattered amplitudes �s remain smaller than 20% and

FIG. 4. Errors on pB (closed sym-

bols) and pNBC (open symbols) as a

function of Nmod in the case of the

scattering by the cylinders C1 (left)

and C2 (right). Decreasing ErNBC-

values at fixed Nmod correspond to

Nit¼ 1, 2, 3, 6, 11. Increasing further

Nmod> 20 and Nit> 11 does not pro-

duce better convergence in these

cases.

FIG. 5. Errors as a function of the

contrast in (left) wavespeed

Dc=~c � 1� c=~c and (right) density

Dq=~q � 1� q=~q. The solid line

shows the amplitude of the scattered

field �s, the errors are ErB (closed

symbols with a dashed line) and

ErNBC (open symbols with a dashed

line for Nit¼ 1 and a dotted line

for Nit¼ 6). Calculations are for

an inclusion located at (x0/H¼ 2.5,y0/H¼ 0.35) for an incident wave in

mode 0 with kH¼ 11.12, 2R/H¼ 0.1.



we observe the same behaviors than when varying the con-

trasts in density and in wavespeed (Fig. 5): ErB � ErNBC Nitð¼ 1Þ � �2

s and the pNBC converges when increasing Nit.

Figure 6(b) shows the same quantities as Fig. 6(a) but

here, the size of the inclusion R has been increased, for fixed kand H, producing both a change in kR and in R/H. With �s < 1,

the same behavior of pNBC is observed, with ErNBC Nitð¼ 1Þ � �2

s and a convergence when increasing Nit. More

importantly, for large R/H-values (here 2R/H> 0.5), the analyt-

ical estimate pB presents a larger error that pNBC(Nit¼ 1), due

to the assumption of small R/H-value used in the former case:

namely ErNBC ’ �2s , whereas ErB ’ �2

s 1þ O R3=H3ð Þð Þ.

D. Limit of validity of the Born approach

As previously said, pB and pNBC are expected to be accu-

rate for weak scattering and we have illustrated that in the

practice, even configurations leading to scattering strengths

�s close to 1 are reasonably described by our approximations.

Figure 7 illustrates the limit of validity of our Born

approach: The same calculations as in Fig. 6(b) have been

conducted but with an higher contrast in wavespeed

c=~c ¼ 0:5, producing a significantly higher scattering, with

�s from a few percent to about 300%.

For small �s-values, until say close to 1 (2R/H< 0.3),

the same trend as previously is observed: ErB � ErNBC Nitð¼ 1Þ � �2

s and iterating the Born approximation causes pNBC

to converge. But we also see that for �s-values larger than

unity, the large error in ErNBC (Nit¼ 1), typically of order

100% causes the solution to diverge when increasing Nit.Similar divergence (not shown) has been observed for

large scattering due to high contrasts in density or wave-

speed or due to large kR-values. In conclusion, the only per-

tinent parameter for the validity of the Born approximation

is the scattering strength and it turns out that it is valid even

for strong scattering (namely for amplitude of the scattered

field close to the amplitude of the incident wave).

IV. ANALYTICAL SOLUTIONS FOR PERIODICALLY ORRANDOMLY DISTRIBUTED INCLUSIONS

In this section, we inspect the analytical solution pB(x, y)

for particular distributions of inclusions: Periodic and random.

In both cases, the solution is compared to results concerning

the scattering by inclusions in the unbounded free space in the

low frequency regime. For periodically distributed inclusions,

particular modes of the waveguides are shown to be excited,

and the equivalence with the usual grating law is exhibited.

Also, the Bragg scattering is recovered. For randomly distrib-

uted inclusions, the reflection and transmission coefficients

are found to account for the shape of the inclusion, particu-

larly relevant in the high frequency regime.

A. Scattering by inclusions periodically distributed,analogy with an infinite array of inclusions

1. Scattering by periodically distributed inclusions

We considered Nx�Ny scatterers periodically distrib-

uted in [0; L]� [0, H] with

x0m ¼ ml; m ¼ 0;…;Nx � 1; l ¼ L=Nx;

y0p ¼ h=2þ ph; p ¼ 0;…;Ny � 1; h ¼ H=Ny: (24)

The sum over the Nx vertical sets of inclusions is accounted

for in Cx ¼ [m

x0m � D; x0m þ D½ �, whereas the sum over the

Ny vertical inclusions appears in cnN; dnN : For each x 2 Cx, a

succession of Ny [ap(x), bp(x)] has to be accounted for thanks

to the transformations cnN ! ScnN and dnN ! SdnN in Eq.

(16) with

FIG. 7. Errors as a function of the size of 2R/H. Same representation as in

Fig. 6 and the errors are calculated for the same configuration, but with an

higher contrast in wavespeed c=~c ¼ 0:5.

FIG. 6. Errors as a function of the

size of the inclusion kR or 2R/H. The

solid line shows the amplitude of the

scattered field �s, and the errors are

ErB (closed symbols with a dashed

line) and ErNBC (open symbols with a

dashed line for Nit¼ 1 and a dotted

line for Nit¼ 6). The errors are calcu-

lated for an inclusion located at (x0/

H¼ 2.5, y0/H¼ 0.35), with no con-

trast in density and a contrast in

wavespeed c=~c ¼ 0:9, for an incident

plane wave (mode 0) with (left) 2R/

H¼ 0.1 varying kR and (right)

kH¼ 11.12 varying 2R/H.



ScnN ¼XNy�1

p¼0

cnN � anNkNHH

2h�1ð Þj1dn Nþ2j1Nyð Þ þ �1ð Þj2dn �Nþ2j2Nyð Þ

h i;

SdnN �XNy�1

p¼0

dnN ¼ nNp2

2knh�1ð Þj1dn Nþ2j1Nyð Þ þ �1ð Þj2dn �Nþ2j2Nyð Þ

h i; (25)

where j1 and j2 are integers. Using Eq. (11), the solution can be written as

p1n xð Þ ¼ eikNxdnN � i q=~q� 1ð Þ 1

H

XNx�1

m¼0

ðmdþD

md�D

dx0 sign x� x0ð ÞScnN þ SdnN½ �� x0ð Þeikn x�x0j jþikNx0

þ ik2 B= ~B� 1� 1

kNknHScnN

XNx�1

m¼0

ðmdþD

md�D

dx0 eikn x�x0j jþikNx0� x0ð Þ þ O �3�

: (26)

It is now sufficient to use

ICx qð Þ¼ 1

H

XNx�1

m¼0

ðmdþD

md�D

dx0 eiqx0� x0ð Þ¼ 1�eiqL

1�eiqLICx0 qð Þ; (27)

where Cx0 is the interval of the first inclusion along the x-axis [�D; þD] [we also consider (1 � eiqL)/(1 � eiql)¼L/l for

q¼ 0], to get the reflection and transmission coefficients by a periodic set of inclusions,

TperionN ¼ dnN þ i � q=~q� 1ð Þ SdnN þ ScnNð Þ þ k2= knkNð Þ B= ~B� 1

� ScnN

� 1� ei kN�knð ÞL

1� ei kN�knð Þl ICx0kN � Knð Þ

RperionN ¼ i � q=~q� 1ð Þ SdnN � ScnNð Þ þ k2= knkNð Þ B= ~B� 1

� ScnN

� 1� ei kNþknð ÞL

1� ei kNþknð Þl ICx0kN � Knð Þ:

8>><>>: (28)

A typical example is given in the following for an incident

wave in mode 1. Figure 8 shows the scattered fields psM and

psB and the corresponding modal components pn (in the pre-

sented case, for an incident wave in mode 1, only odd modes

are excited).

The transmission coefficients Tn1 deduced from MELINA

calculation for even modes are of order 10�3 (our prediction

is zero for these modes). The transmission coefficients for

odd modes are TM11 ¼ �1:24i and TM

31 ¼ 0:20þ 0:26i in good

agreement with our predictions from Eq. (28): TM11 ¼ �1:29i

and T31¼ 0.24þ 0.25i.

2. Analogy with an infinite array of inclusions, gratinglaw and Wood’s anomalies

For an infinite array of h-spaced scatterers, the classical

grating criterion gives the directions hn in which constructive

interferences are observed:

kh sin hm ¼ kh sin hþ 2mp; (29)

where h is the angle of the incident wave (measured from

the x-axis). The symmetry of the problem and the Neumann

boundary conditions on the waveguide walls makes the con-

figuration analogous to an infinite array of inclusions for an

incident wave formed of the superposition of two waves

with incidences 6h satisfying k sin h¼Np/h. Note that gen-

eral incidences can be treated in the context of Rayleigh

Bloch waves leading to pseudo-periodic boundary conditions

(instead of Neumann boundary conditions).34 Here, the angle

hm has to be related to an order n of a propagating mode,

namely np : kh sin hm. The grating criterion, Eq. (29), gives

that n and N have the same parity, n¼Nþ 2m, in agreement

with the selection of propagating modes in Eq. (25) [with

Ny¼ 1 in Eq. (25)]. Obviously, the selection of evanescent

modes in Eq. (25) is not accounted in the grating criterion

(then jsin hmj> 1), which considers only the far field.

Finally, it appears that the representation in Eq. (26)

breaks down if kn¼ 0 (cos hm¼ 0), a singularity known as

Wood’s (or Rayleigh) anomaly.35,36 It corresponds to a spec-

tral order emerging at grazing angle, or to a mode passing a

cutoff frequency. This has been analyzed in details in the

case of sound hard and sound soft cylinders in Ref. 37.

3. The case of an incident plane wave, comparisonwith Twersky’s prediction

Although our prediction enables us to consider an inci-

dent wave forming an angle h with the array of inclusions,

we consider for simplicity the case of an incident plane wave

and we consider a grating with Ny¼ 1. In that case, the equa-

tions in Eq. (28) simplify to



Tperion0 ¼ dn0 þ ian0

kS

2H� q=~q� 1ð Þ þ k=kn B= ~B� 1

� �cos

np2S k � knð ÞRð Þ

Rperion0 ¼ ian0

kS

2Hq=~q� 1ð Þ þ k=kn B= ~B� 1

� �cos

np2S k � knð ÞRð Þ

8><>: (30)

with S¼pR2 the surface of the inclusion and

S qRð Þ � 2J1 qRð ÞqR

(31)

a factor depending on the shape of the inclusion (here given

for a cylindrical shape). These predictions are expected to be

valid for small R/H-values, but they are valid even in the

high frequency regime kR, kH> 1, where several propaga-

tive modes exist, corresponding to several orders of interfer-

ence as previously commented. Typical scattered fields are

presented in Fig. 9(a) in the low frequency (kR¼ 0.7,

kH¼ 14, thus 4 propagative modes) and in the high fre-

quency (kR¼ 1.8, kH¼ 36, thus 11 propagative modes)

regimes. Figure 9(b) shows the corresponding reflection

coefficients as a function of the n-value. As expected from

the symmetry of the system, only the even modes are

excited. This condition is encapsulated in the term cos np/2

in Eq. (30). It also corresponds to the grating criterion

Eq. (29), k¼ 2mp/H (and n¼ 2m). We observe a good agree-

ment between direct calculation of Rn0 (close symbols) and

our analytical prediction [open symbols, from Eq. (30)].

Incidentally, in the high frequency regime (kH¼ 36),

where the field pattern can be analyzed in terms of ray trajec-

tory, the observed pattern (here in reflection) clearly corre-

sponds to the superposition of two plane waves forming an

angle of, respectively, 6 h, with tan h� 1.8. This h-value

corresponds to the mode n¼ 10, using tan h¼ ky/kn,

[ky¼ np/H, kn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � np=Hð Þ2

q]. Here, the kH-value is

close to the cutoff frequency of the mode 10 at kcH¼ 31.4,

and we have seen that a mode is resonant at its cutoff fre-

quency (Wood’s anomaly).

In Fig. 10, we reported the variation of jRn0j varying the

frequency kR (same configuration as in Fig. 9 has been con-

sidered). Results deduced from MELINA calculations and our

estimate Eq. (30) (solid line) are in good agreement (the

errors ErB in the whole range of kR-values is less than 8%).

Figure 10 also illustrates the pertinence of the shape factor S

FIG. 8. (Color online) Scattering by a periodic

set of scatterers [x0p/H¼ (2, 2.5, 3), y0m/

H¼ (0.25, 0.75)] for the mode 1 incident

(k¼ 5.56, R¼ 0.05, c=~c ¼ 0:9, q ¼ ~q). (a) Scat-

tered field psM, on the bottom ps

B using Eqs.

(4.5). The error is ErB¼ 6%. (b) pn(x) for

n¼ 0,.,6, solid lines correspond to the results

obtained with MELINA code, dashed–dotted line

to the solution using Eq. (4.5). Only odd modes

are expected to be nonvanishing from Eq. (4.2).



in Eq. (30) when increasing the frequency (dotted line give

the result when omitting this factor).

It is of interest to compare our result with the prediction

given by Twersky38,39 in the low frequency regime,

namely the limit kR; kH ! 0. With f(h) the scattering func-

tion of a single inclusion in free space [here defined as

ps rð Þ ! f hð Þffiffiffiffiffiffiffiffiffiffiffiffi2=pkr

peikr�ip=4 in the far field], he found (a

clear derivation can be found in Eq. (3.1.a,b) of Ref. 40],

TT¼1þ4i f 0ð Þ2ikH

¼1þ ikS

2H�2

q� ~qqþ ~q

þ B= ~B�1� � �

RT¼4i f pð Þ2ikH

¼ ikS

2H2q� ~qqþ ~q

þ B= ~B�1� � �

;

8>><>>: (32)

where we used, for cylindrical inclusions 4i f hð Þ ¼ p kRð Þ21� B= ~B �

þ 2p kRð Þ2 q� ~qð Þ= qþ ~qð Þ cos h.

FIG. 10. Reflection coefficients jRn0j as a function of the

frequency kR (same configuration as in Fig. 9 has been

considered). Open circles from MELINA calculations, solid

lines from our prediction Eq. (4.7), and dashed line using

Eq. (4.7) but omitting the factor of shape S. Vertical lines

indicate some cutoff frequencies from kH¼ 2p (kR¼ 0.31)

to kH¼ 10p (kR¼ 1.57).

FIG. 9. (Color online) (a) Scattered fields

psM x; yð Þ by an infinite array of cylindrical inclu-

sions H-spaced and (b) corresponding reflection

coefficients jRn0j as a function of n. The inclu-

sion has a contrast in wavespeed c=~c ¼ 0:9 and

2R=H ¼ 0:1 (no contrast in density). (Top)

kR¼ 0.7 and (bottom) kR¼ 1.8.



This coincides with the plane wave approximation,

where our estimates in Eq. (30) simplify in

Tperio00 ¼ 1þ i

kS

2H� q=~q� 1ð Þ þ B= ~B� 1

� �Rperio

00 ¼ ikS

2Hq=~q� 1ð Þ þ B= ~B� 1

� �S 2kRð Þ:

8><>: (33)

There are obviously differences between the predictions: (1)

Our predictions account for the shape of the inclusion and

for the frequency regime through the term S(2kR) and (2)

the dependence in the contrast in density in Twersky’s pre-

diction differs from ours, except for small contrasts. These

two discrepancies are inspected in the following.

We calculated from MELINA results the reflection coeffi-

cient in the low frequency regime (kR¼ 0.15, kH¼ 3) for

various contrasts in density and for various contrasts in

wavespeed. The results are reported in Fig. 11. In the consid-

ered configuration, S(2kR)¼ 0.99 is irrelevant and our esti-

mate on Rperio00 differs from RT only on the dependence on the

contrast in density. Both predictions appear to be accurate

when varying the contrast in wavespeed. When varying the

contrast in density, our estimate (solid line, MELINA results

are given by open circles) appears to be less accurate than

Twersky’s (dotted line). This means that the first Born

approximation gives the leading order contribution q=~q.

Note that this has already been observed for the scattering of

spheres in free space.25 Then, iterating the NBC (star sym-

bols in Fig. 11) recovers the dependence 2 q� ~qð Þ= qþ ~qð Þthat is approximated by q=~q� 1ð Þ in the first iteration. It fol-

lows a significant limitation of our analytical prediction

unable to account for sound hard inclusions. We conclude

that the first Born estimate is less accurate in the low fre-

quency regime than other perturbative approximations (a

conclusion already observed in Ref. 25) and attaining better

accuracy requires to iterate NBC. On the other hand, it

allows us to go beyond the low frequency regime, and to

account for the shape of the inclusions. Further research is

needed to inspect that possibility.

B. Random distribution of inclusions

1. The case of randomly distributed inclusions in awaveguide

The case of a random distribution of Ns small inclusions

occupying the space xi 2 0; L½ �, yi 2 0;H½ �, i ¼ 1;…;Ns is

straightforward considering the linearity of Eq. (13). We

have

p1n xs < x< 0ð Þ ¼ eikNxdnN þ

XNs

i¼1

RnN xi;yið Þe�iknx

p1n x> Lð Þ ¼

XNs

i¼1

TnN xi;yið Þeiknx

8>>>><>>>>:

(34)

with

TnN xi; yið Þ ¼ dnN þ i � q=~q� 1ð Þ dnN yið Þ þ cnN yið Þð Þ þ k2= knkNð Þ B= ~B� 1�

cnN yið Þ �

ICxikN � knð Þ

RnN xi; yið Þ ¼ i � q=~q� 1ð Þ dnN yið Þ � cnN yið Þð Þ þ k2= knkNð Þ B= ~B� 1�

cnN yið Þ �

ICxi kN þ knð Þ;

((35)

and

ICxiqð Þ�1

H

ðCxi

dx0� x0ð Þeiqx0 ; (36)

The mean field is obtained by averaging over all realizations of disorder, and if we assume no correlations between scatterers

�h i � 1

HNs

1

LNs

ðdx1 � � � dxNs

ðdy1 � � � dyNs

: (37)

The average over the xi concerns ICxonly and the average over the yi concerns cnN and dnN. We get

FIG. 11. Reflection coefficient as a function of the contrast in density q=~qand as a function of the contrast in wavespeed c=~c in the low frequency re-

gime (kR¼ 0.15, kH¼ 3). Open circles, deduced from MELINA calculations

(the point at q=~q ¼ 0 has been obtained by applying Neumann boundary

conditions at the inclusion boundary). Solid lines from our estimate in Eq.

(33) and dotted lines from Twersky’s estimate Eq. (32). Star symbols show

the reflection coefficient when using the results of NBC with Nit¼ 10.



TrandNN ¼ 1þ iu

2k=kNð Þ2 B= ~B� 1

� � q=~q� 1ð Þ 1þ Npð Þ2= kNHð Þ2

� h ikNL

RrandNN ¼ �

u4

B= ~B� 1�

k=kNð Þ2þ q=~q� 1ð Þ 1� Npð Þ2= kNHð Þ2� h i

1� e2ikN L�

S 2kN Rð Þ

8><>: (38)

and RrandnN ¼ Trand

nN ¼ 0 if n 6¼ N. We have defined u � NsS=LH the volume fraction of the inclusions.

Some remarks can be made at this stage: First, no mode

conversion occurs, a result that has been already observed for

hard inclusions from direct numerical calculations,41 also

observed in different context.42,43 The dependence of TrandNN on

the length L of the slab containing the inclusions is linear, a

consequence of the linearity of our equation in the single scat-

tering approximation, which causes the prediction on the trans-

mission unreliable for large L. The limit of TrandNN for a large

slab is less problematic: Including a small attenuation in the

host medium that produces a small imaginary part in kN, and

the limit L!1 becomes possible [namely e2ikNL vanishes in

Eq. (38)]. This limit will be inspected in the following section.

Numerical experiments to inspect the characteristics of

effective media is quite difficult and only few attempts have

been made.41,44 It is out of the scope of the present paper to

exhaustively study these characteristics as a function of the

multiple parameters of the problem. We give a preliminary

result concerning a unique configuration. Inclusions with

kR¼ 0.9, 2R/H¼ 0.1 are randomly distributed in a slab of

length L/H¼ 3, the inclusion filling fraction is u ¼ 10%. A

typical realization of this disordered configuration is shown

in Fig. 12, where the solutions pM and pB appear to be in

good agreement, with an error ErB¼ 4% [the modal compo-

nents are also compared in Fig. 12(b)]. Here, pB is calculated

by summing the fields for each inclusion [Eqs. (13)–(17)],

which produces an error as only the field reflected and trans-

mitted by an inclusion is available analytically.

Averaging over several realizations of the disorder is

expected to produce a coherent effective wave. In free space,

an usual modelization of the resulting coherent field is

expressed as ph i x; yð Þ / eiKx, where K has a real part that

encapsulates the change in the wave speed due to the pres-

ence of the inclusions and an imaginary part due to the de-

structive interferences produced in the average process. The

study of the validity of such modelization for guided propa-

gation and the links between effective propagation in free

space and effective guided propagation will be the subject of

future works. Here, we propose only to test the validity of

the transmission and reflection coefficients.

Figure 13 shows the mean scattered field psM

� �x; yð Þ

obtained using MELINA code after 1000 realizations of the dis-

order (the disorder corresponds here to the random positions

of the inclusions, otherwise considered identical in size and

contrast). The total field ph i ¼ p0 þ psM

� �is not reported

because the mean scattered field, with 20% in amplitude

with respect to the incident wave, is too weak to produce a

visible change (only the incident plane wave is visible). Let

us however stress that the scattered mean field has an

increasing amplitude when it “propagates” in the waveguide,

a fact that is not intuitive, whereas the total mean field has a

decreasing amplitude as expected, because of destructive

interferences between p0 and psh i.Inspecting this effective mean field gives the following

conclusions:

(i) Figure 13(b) shows the modal components pnh i xð Þ of

psM

� �xð Þ for n¼ 1–4, to be compared with the modal

components obtained for one realization [Fig. 12(b)].

In agreement with our prediction, the average causes

the mode conversions to vanish and only mode 0 sur-

vives. Numerically, the amplitudes of the higher

modes decrease in the average process and their maxi-

mum value after a finite number of averages can be

used to define the limit of resolution in psM

� �. Here,

with 1000 averages, the maximum amplitude is

observed for n¼ 1 and equals 0.02 (2% of the total

mean field for a mode 0 corresponding to about 20%

of the total mean field).

(ii) The transmission coefficient Trand00 can be deduced

from p0h i x=H > 4ð Þ ¼ Trand00 eikx (in each configura-

tion, the inclusions are contained in the slab

x=H 2 1; 4½ �): we get Trand00 ¼ 1:04� 0:255 i, in

FIG. 12. (Color online) (a) Scattered field calcu-

lated with MELINA psM and ps

B (resulting error is

ErB¼ 4%) and (b) corresponding modal compo-

nents pn(x) for n¼ 0–3; solid line calculated from

MELINA psM and dotted line pB,n. All the inclusions

(here, 38 inclusions) have the same characteristics

(2R=H ¼ 0:1, q=~q ¼ 1, c=~c ¼ 0:95) and are

located randomly between x/H¼ 1 and x/H¼ 4

with a filling fraction /¼ 10%. The incident

wave is a plane wave (mode 0) with kR¼ 0.9.



reasonable agreement with our estimate Eq. (38)

Trand00 ¼ 1� 0:263 i (the error bar being 0.02 as previ-

ously discussed). The reflection coefficient can be

deduced from p0h i x=H < 1ð Þ ¼ Rrand00 e�ikx. Numeri-

cally, we find a value of order 10�3 (our prediction

gives Rrand00 ¼ 0:001� 0:0015 i), but the values cannot

be compared quantitatively because they are well

below the resolution.

2. The case of a single interface, comparison withunbounded effective propagation

Our result concerns the reflection and transmission coeffi-

cients through a slab of randomly located inclusions in a

waveguide. As previously said, this should allow us to deduce

the characteristics of the guided effective wave, if it makes

sense. Also, comparisons with results available for unbounded

effective propagation make sense: Does a finite slab of ran-

domly located inclusions behave as an unbounded random

distribution? To answer this question, more precise and sys-

tematic numerical experiments are needed and this will be the

subject of future works. However, a hint of how both guided

and unbounded situations compare is the case of a unique

interface between the free space and the effective medium. In

that case, there is a prediction for the reflection coefficient at

the interface in the low frequency regime45–47 (small cylindri-

cal inclusions with kR� 1), RM [Eq. (2) in Ref. 47],

RM ¼ iuf pð Þk2S¼ �u

4B= ~B� 1�

þ 2q� ~qqþ ~q

� �; (39)

where f is, as in the periodic case, the scattering function.

Considering the limit L!1 in our Eq. (38), we get

Rrand00 ¼ �u

4B= ~B� 1�

þ q=~q� 1ð Þ �

S 2kRð Þ; (40)

where we have used the artifact of a small attenuation in the

host medium to take the limit eikL! 0. Considering the limit

ka ! 0 (as assumed in Refs. 46 and 47) and q � ~q causes

the two results to coincide, which confirms the validity of

the effective medium approach for guided waves. It is how-

ever of interest for future work to test the validity of our

result for any ka value (recall indeed that we assumed

a=H � 1 but not ka� 1 a priori).

V. CONCLUSIONS

We studied the propagation of acoustic waves in a

waveguide containing inclusions of any shape and presenting

contrasts in wavespeed and in density with respect to the

host medium. An iterative modal approach is developed.

This allows us to get a numerical scheme (NBC), easy to

implement due to implying one-dimensional quantities (the

modal components of the two-dimensional field). By com-

parison with numerical experiments (performed using a finite

element method on MELINA code), the method based on the

Born approximations is shown to be accurate, even for scat-

tered fields whose amplitudes are of the same order of mag-

nitude as that of the incident wave. This scattered amplitude

(�s) is the relevant parameter for the accuracy of the method,

typically accurate in O �2s

� in the first iteration. Until values

of �s close to unity, iterating the NBC produces a conver-

gence of the solution. The �s-value results from the com-

bined effects of the inclusion size and of the contrasts in

density and in wavespeed. This means that no individual

limitations applies: Namely, a large inclusion with weak

contrasts can produce a small �s-value. Note also that �s can

be evaluated, in practice, by considering the scattering

strength of the same inclusion, or set of inclusions, in free

space (this latter quantity is, in general, easier to obtain).

For inclusion sizes R, small compared to the waveguide

height H, analytical expressions of the transmission and

reflection coefficients are given. Because the restriction con-

cerns the R/H-value only, our predictions hold for any

kR-value, which is of particular interest to explore the high

frequency regime. This is encapsulated in a factor S(kR),

FIG. 13. (Color online) (Top) Mean scattered

field psM

� �x; yð Þ obtained after averaging 1000

configurations of the randomly located inclu-

sions. (Bottom) Corresponding mean modal

components pnh i xð Þ for n¼ 0–3. For each real-

ization of the disorder, the characteristics of the

inclusions (size and contrasts) are the same as

in Fig. 12 but the positions change.



that accounts for the shape of the inclusion and for the fre-

quency regime [with S kRð Þ ! 1 for kR! 0]. Comparison

with numerical experiments shows that the results is accurate

typically in O �2s

� Oð R=Hð Þ2Þ.

The scattering properties of periodically and randomly dis-

tributed inclusions have been considered in more detail. The

main conclusions are the following. Our approach, being per-

turbative, is unable to account for resonance effects. This

appears in the divergence of pB near the cutoff frequencies, or

equivalently for periodic case in the Wood’s anomalies. When

compared to other perturbative approaches valid in the low fre-

quency regime, our analytical predictions are less accurate at

low frequencies, although iterating the Born approximation in

that case causes the solution to converge to the correct solution.

In addition, the main advantage of our approach is the validity

of our predictions in the high frequency regime.

Future works concern the case of randomly distributed

inclusions. By comparing with numerical experiments, it will

be possible to explore the bounds of validity of our prediction

beyond the low frequency regime. Extension to elasticity is

possible, by considering modal expansion on the Lamb modes.

ACKNOWLEDGMENT

This work is supported by the international ANR project

Procomedia.

APPENDIX: NUMERICAL IMPLEMENTATION OF THESUCCESSIVE NBC

pNBC(x, y) is calculated in the successive NBC using the

Nit-th iteration in Eq. (11) then

pNBC x; yð Þ ¼XNmod

n�0

pNitn xð Þwn yð Þ; (A1)

which depends on (Nit, Nmod) and is expected to converge when

increasing both parameters. The numerical procedure to derive

pNitn xð Þ is straightforward. We define A1 � q=~q� 1ð Þ=H and

A2 � k2 B= ~B� 1�

=H for simplicity. Initialization is given

with pð1Þn xið Þ ¼ eikNxidnN and pð1Þ0n xið Þ ¼ ikNeikNxidnN . Then,

with

N1ð Þ

1 j; nð Þ �X

m

A1Cnm xj

� p 1ð Þ0

m xj

� ;

Nð1Þ2 j; nð Þ �

Xm

A1Dnm xj

� � A2Cnm xj

� �p 1ð Þ

m xj

� (A2)

and a parametrization of the inclusion located between x1

and x2, xi¼ x1þ i Dx with i¼ 0,...,Nx and Dx¼ (x2� x1)/Nx,

p 2ð Þn xið Þ ¼ eikNxidnN þDx

Xj

�g0n xi� xj

� N

1ð Þ1 j;nð Þ

h

þgn xi� xj

� N

1ð Þ2 j;nð Þ

i;

p 2ð Þ0n xið Þ ¼ ikNeikN xidnN �N

1ð Þ1 i;nð Þ

þDxX

j

k2ngn xi� xj

� N

1ð Þ1 j;nð Þ

h

þg0n xi� xj

� N

1ð Þ2 j;nð Þ

i: (A3)

Iteration is done using pð1Þn xið Þ ¼ pð2Þn xið Þ and pð1Þ0n xið Þ¼ pð2Þ0n xið Þ. The final x-mesh Xi ¼ iL=Nx, i ¼ 0;…;Nx is

used after pNit�1n xj

� has been calculated

p Nitð Þn Xið Þ ¼ eikNxidnN

þDxX

j

�g0n Xi� xj

� N

Nit�1ð Þ1 j;nð Þ

h

þgn Xi� xj

� N

Nit�1ð Þ2 j;nð Þ

i: (A4)

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